Pith. sign in

REVIEW 3 major objections 4 minor 24 references

Special anisotropic conformal changes of conic pseudo-Finsler surfaces

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read An anisotropic conformal change of a conic pseudo-Finsler surface satisfies the vertical $\varphi T$-condition exactly when the original metric has vanishing $T$-tensor, so every Landsberg surface changed this way is Berwaldian.

desk verdict A correct but derivative extension of the authors' anisotropic conformal framework; the vertical φT-condition equivalence is clean, but the paper rests on unpublished formulas and contains a false projective-flatness claim. read the letter →

arxiv 2505.22593 v1 pith:7HW4YIS3 submitted 2025-05-28 math.DG

Nabil L. Youssef , Ebtsam H. Taha , A. A. Kotb , S. G. Elgendi This is my paper
classification math.DG MSC 53B4053C60
keywords anisotropicconformalchangeconicpseudo-FinslersurfacemodifiedBerwaldframeC-anisotropicφT-conditionT-tensorLandsbergFinslerianSchwarzschild-deSitter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies rescaling a conic pseudo-Finsler surface—a surface with a smooth, direction-dependent metric that may be indefinite—by a factor depending on both position and direction. Because such a change no longer leaves the Cartan tensor or the $T$-tensor invariant, the classical $C$-conformal and $\sigma T$-conditions split into several distinct variants. The paper characterizes each variant: the $C$-anisotropic and horizontal $C$-anisotropic changes reduce to the usual $C$-conformal change when the factor depends only on position, while the vertical $C$-anisotropic change holds exactly when the underlying metric is Riemannian. Its central discovery is that the vertical $\varphi T$-condition—contraction of the $T$-tensor with the vertical derivative of the conformal factor—is equivalent to the original metric having vanishing $T$-tensor, so any Landsberg surface satisfying it is automatically Berwaldian. An explicit Finslerian Schwarzschild-de Sitter sphere obtained by such a rescaling of a Riemannian sphere illustrates the apparatus.

What carries the argument

The machinery is the modified Berwald frame $(\ell_i,m_i)$ for a two-dimensional Finsler surface, together with the v-scalar and h-scalar derivatives $f_{;1}, f_{;2}, f_{,1}, f_{,2}$. In this frame the Cartan tensor has the form $(I/F)m_im_jm_k$ and the $T$-tensor has the form $(I_{;2}/F)m_im_jm_km_r$, so contractions reduce to scalar conditions. Under the change $\bar{F}=e^{\varphi}F$, the frame and main scalar transform through formulas involving $\rho=1/(\sigma+\varepsilon-(\varphi_{;2})^2)$, and the paper imports transformation formulas for $I_{;2}, I_{,1}, I_{,2}$ from a companion preprint to compute the transformed $T$-tensor and its contractions.

What would settle it

Take a non-Riemannian Finsler surface with non-vanishing $T$-tensor and a non-isotropic smooth function $\varphi$, then compute $(\dot{\partial}_i\varphi)T^{i}_{jkr}=0$ in the modified Berwald frame; Lemma 4.11 predicts the equation reduces to $\varphi_{;2}I_{;2}=0$, so exhibiting a solution with $\varphi_{;2}\neq 0$ and $I_{;2}\neq 0$ would contradict it. A direct coordinate evaluation of formula (4.2) for the transformed $T$-tensor on such a surface would settle the same question.

Watch

Extended reading notes

Core claim

The paper's central claim, stated in Lemma 4.11 and Remark 4.12, is that for the anisotropic conformal change $\bar{F}=e^{\varphi}F$ on a conic pseudo-Finsler surface, the vertical $\varphi T$-condition $(\dot{\partial}_i\varphi)T^{i}_{jkr}=0$ holds if and only if $F$ has vanishing $T$-tensor. Consequently the vertical $\varphi T$-condition is equivalent to the $T$-condition, and every Landsberg surface that undergoes such a change is Berwaldian. The paper also shows that vertical $C$-anisotropic changes, defined by $(\dot{\partial}_i\varphi)C^{i}_{jk}=0$, are characterized by $F$ being Riemannian. These results are obtained by expressing every tensor contraction in the modified Berwald frame, where the Cartan and $T$-tensors reduce to scalar multiples of products of the frame vector $m_i$, so each condition becomes a scalar equation in the main scalar $I$ and the conformal factor's derivatives.

Load-bearing premise

The central equivalences in Section 4 rest on transformation formulas quoted from the authors' companion preprint, together with the assertion that $\rho$ and $\rho_{;2}$ are horizontally constant whenever $\varphi_{;2}$ is; an error in any of these identities would undermine the equivalence claims.

Editorial extensions

If this is right

  • If an anisotropic conformal change on a conic pseudo-Finsler surface satisfies the vertical $\varphi T$-condition, the original metric has vanishing $T$-tensor, and any Landsberg surface changed this way is Berwaldian.
  • Vertical $C$-anisotropic conformal changes exist only when the starting metric is Riemannian, so they cannot turn a genuinely Finsler surface into a different genuinely Finsler surface.
  • For a position-dependent conformal factor, the $C$-anisotropic and $\overline{C}$-anisotropic conditions, together with their horizontal versions, all reduce to the classical $C$-conformal condition, while the vertical $C$-anisotropic condition does not.
  • When the conformal factor is position-only, the $\varphi T$, $\overline{\varphi}T$, horizontal $\varphi T$ and horizontal $\overline{\varphi}T$ conditions reduce to the $\sigma T$-condition, but the vertical $\varphi T$-condition instead becomes the $T$-condition.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the vertical $\varphi T$-condition is equivalent to vanishing $T$-tensor, it is a property of the original metric rather than of the chosen rescaling; this offers a way to test for $T$-vanishing by constructing one anisotropic conformal factor and checking a vertical contraction, instead of computing the full $T$-tensor.
  • The same scalar-frame technique may apply to other non-Riemannian tensors: imposing vertical contraction conditions with the Landsberg tensor or Berwald curvature could yield similarly strong rigidity statements for Landsberg or weak-Berwald surfaces, though the paper does not investigate this.
  • The Schwarzschild-de Sitter construction suggests a general recipe for producing Finslerian spacetimes from Riemannian seeds by anisotropic conformal rescaling; verifying that such metrics solve Finslerian Einstein equations in four dimensions would require extending the surface-level scalar conditions beyond two dimensions.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies special anisotropic conformal changes F = e^{φ} F of conic pseudo-Finsler surfaces, with φ depending on position and direction. It introduces and characterizes C-anisotropic, horizontal C-anisotropic, and vertical C-anisotropic conformal changes (Definitions 3.3, 3.8, 3.13), and similarly defines φT-, horizontal φT-, and vertical φT-conditions (Definitions 4.3, 4.7, 4.10). The central result is Lemma 4.11: a proper anisotropic conformal change satisfies the vertical φT-condition if and only if F has vanishing T-tensor, from which the paper concludes that every Landsberg surface satisfying this condition is Berwaldian and that the vertical φT-condition is equivalent to the T-condition. The final section applies the framework to a Randers-type metric on the two-sphere, presented as a Finslerian Schwarzschild-de Sitter solution.

Significance. If the claims are correct, the paper gives a useful organizing classification of anisotropic conformal changes and their interaction with the Cartan tensor and T-tensor on Finsler surfaces. The proof of Lemma 4.11 is direct, self-contained, and does not depend on the unpublished transformation formulas quoted from [23], which is a genuine strength of the central claim. The paper also provides a concrete example with a Maple verification link, which is helpful for reproducibility. However, the significance is tempered by the fact that several peripheral but published claims rely on identities from an unpublished preprint, and by at least one false assertion about the round sphere in the example section.

major comments (3)
  1. [§6, item (i)] The assertion that the round Riemannian metric F on S^2 is not locally projectively flat is false. A two-dimensional Riemannian metric of constant sectional curvature is locally projectively flat by Beltrami's theorem; the round sphere admits local coordinates, e.g., gnomonic coordinates, in which geodesics are straight lines. The failure of Hamel's equation in the chosen (θ, η) coordinates only shows that those coordinates are not projective coordinates; it is not a coordinate-invariant obstruction. Please correct this statement or replace it with a correct coordinate-invariant argument.
  2. [§4, Eqs. (4.2)–(4.8), Theorem 4.9] Several load-bearing identities are quoted without proof from the unpublished preprint [23], including the transformed T-tensor formula (4.2), the scalar derivative formulas (4.3)–(4.8), and the results [23, Corollary 2.5 and Proposition 4.7] used in the proof of Theorem 4.9. Because the barred versions of the φT-conditions and the horizontal/vertical equivalences in Theorem 4.9 and Corollary 4.13 depend on these identities, the manuscript is not self-contained. Please either include full proofs of these identities or clearly state them as assumptions from the companion preprint; alternatively, restrict the claimed equivalences to the unbarred vertical φT-condition, whose proof in Lemma 4.11 is self-contained.
  3. [§7, Table] The concluding table contradicts the body of the paper. The row for 'vertical φT' states 'F is Riemannian', but Lemma 4.11 proves that the vertical φT-condition is equivalent to F having a vanishing T-tensor, which is a strictly weaker condition. In addition, the rows for 'φT-condition' and 'φT-condition' are identical, whereas Lemma 4.4 gives different alternatives for the two cases: mi∂iφ = 0 versus mi∂iφ − εφ;2ℓi∂iφ = 0. The table should be corrected and the barred/unbarred notation made unambiguous in every row.
minor comments (4)
  1. [§6] The numbered list in Section 6 jumps from item (v) to item (vii), skipping (vi); renumber the items.
  2. [Remark 4.12] The statement 'The φT-condition is equivalent to the T-condition' should specify that this is the vertical φT-condition, and the bar notation for the companion condition should be used consistently so that the reader can distinguish φT from φT.
  3. [Proposition 3.10] The condition 'either φ,2 = 0 or φ,1 = 0' is imprecise: under the preceding horizontal C-anisotropic condition φ,2 = φ;2φ,1 with φ;2 ≠ 0, either alternative forces both φ,1 and φ,2 to vanish. The statement should read 'φ,1 = φ,2 = 0'.
  4. [Definition 2.2] The nondegeneracy condition in display (2.8) is written densely as 'F^2(∂̇i∂̇jφ + (∂̇iφ)(∂̇jφ))m^i m^j + ε = ε + σ − (φ;2)^2 ≠ 0'; separating the transformation law from the nondegeneracy condition would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: Lemma 4.11 derives the vertical φT-condition ⇔ T-condition equivalence directly from the displayed frame identities, so the headline claim is self-contained; the authors' unpublished preprint [23] is load-bearing only for the secondary Theorem 4.9, which is a verification gap rather than a circular step.

full rationale

The central claim of Section 4 — that the vertical φT-condition is equivalent to the T-condition (Lemma 4.11, Remark 4.12, Corollary 4.13) — is self-contained. With F T^i_{jkr} = I;2 m^i m_j m_k m_r and (2.4), the contraction (∂̇_i φ)T^i_{jkr} = (I;2/F)(∂̇_i φ)m^i m_j m_k m_r = ε I;2 φ;2 F^{-2} m_j m_k m_r vanishes for a proper change (φ;2 ≠ 0) exactly when I;2 = 0; only the displayed identities (2.2)-(2.4) are used, so no quoted formula from the authors' prior work enters. The Landsberg-to-Berwald conclusion then follows from the displayed commutation formula (2.6): I,1 = 0 and I;2 = 0 force I,2 = 0. The new conditions are natural analogues of the σT-condition (∂_i σ)T^i_{jkr} = 0 from [5], and the equivalences are one-line derivations rather than definitions of the conclusions; no parameter is fitted, and no uniqueness theorem is imported. The paper does, however, rely heavily on the authors' own work: transformation formulas (2.10)-(2.15) are from the published [22], and formulas (4.2)-(4.8) together with [23, Corollary 2.5] and [23, Proposition 4.7] are taken from the same authors' unpublished preprint [23]. This reliance is load-bearing for Theorem 4.9, whose proof asserts '(i) ⇔ (ii) It follows by [23, Proposition 4.7] and Proposition 4.8 (ii)', and for the barred halves of Lemmas 4.4 and 4.8; if those unverified identities failed, Theorem 4.9 would collapse. That is a verification gap and a correctness risk for a secondary result, but not a circular reduction, since the argument chain does not use the conclusions to prove themselves. Two internal inconsistencies are flagged: the summary table's row 'vertical φT: F is Riemannian' contradicts Lemma 4.11 (vanishing T-tensor), and its barred φT-condition row duplicates the unbarred row; these are typographical and do not infect the theorems.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No numerical parameters are fitted in the main theorems; the arbitrary h(0) function phi is the object of study rather than a hidden fitted constant. The main load is carried by the anisotropic conformal machinery of [22] and especially by unproved identities from the unpublished preprint [23]. The only hand-chosen constant is the Finsler parameter a in the worked example.

free parameters (1)
  • Finsler parameter a = a (0 <= a < 1)
    Chosen by hand in the Section 6 example to define the Finslerian Schwarzschild-de Sitter metric; it is not fitted to data and is not part of the main theorems.
assumptions (5)
  • standard math The modified Berwald frame and scalar derivatives for two-dimensional conic pseudo-Finsler surfaces satisfy equations (2.1)-(2.7).
    Background from Basco-Matsumoto and standard Finsler surface theory, used throughout Sections 2 to 4.
  • domain assumption The anisotropic conformal change (2.8) and the frame and main scalar transformation formulas (2.10)-(2.15) from [22] are valid.
    Taken from the authors' published paper [22]; no proof is reproduced here.
  • domain assumption The transformed T-tensor and scalar derivative formulas (4.2)-(4.8), and the statement that rho and rho;2 are horizontally constant used in Theorem 4.9, are correct as stated in [23].
    [23] is an unpublished preprint by the same authors; these formulas are load-bearing for the phi T-condition results.
  • standard math A Landsberg Finsler surface with vanishing T-tensor is Berwaldian.
    Known fact cited from [5], used in Theorems 4.5 and Corollary 4.13.
  • standard math Projective flatness for a Finsler surface is characterized by Hamel's equation in adapted coordinates, as stated in Remark 3.5.
    Used in Proposition 3.6 and Section 6; Section 6 applies it in a way that appears to confuse coordinate-dependent Hamel failure with non-flatness.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Special anisotropic conformal changes of conic pseudo-Finsler surfaces." pith.science (2026). https://pith.science/paper/7HW4YIS3

@misc{pith2026250522593,
  author       = {Pith},
  title        = {Pith review of: Special anisotropic conformal changes of conic pseudo-Finsler surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7HW4YIS3}},
  note         = {Machine review of arXiv:2505.22593}
}
abstract

This study presents many special anisotropic conformal changes of a conic pseudo-Finsler surface $(M,F)$, such as $C$-anisotropic and horizontal $C$-anisotropic conformal transformations, which reduce to $C$-conformal when the conformal factor is solely position-dependent. Furthermore, we present vertical $C$-anisotropic conformal changes and demonstrate that they are characterized by the property of $(M,F)$ being Riemannian. Additionally, we examine the anisotropic conformal transformation that fulfils the $\phi T$-condition, the horizontal $\phi T$-condition, and the vertical $\phi T$-condition. The first two conditions reduce to the $\boldsymbol{\sigma} T$-condition when the conformal factor relies solely on a positional variable. We demonstrate that, under the vertical $\phi T$-condition change, every Landsberg surface is Berwaldian. Thus, the vertical $\phi T$-condition is equivalent to the $T$-condition. Furthermore, we examine the scenario when the anisotropic conformal factor becomes the main scalar of the non-Riemannian surface $(M,F)$. We present an example of a Finslerian Schwarzschild-de Sitter solution having Finslerian spherical symmetry and apply our results to it.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [23]

    N. L. Youssef, S. G. Elgendi, A. A. Kotb and E. H. Taha, Anisotropic Conformal Change of Conic pseudo-Finsler Surfaces II , arXiv:2503.07842v1 [math.DG], Submitted

    work page Pith review arXiv

  2. [1]

    P. L. Antonelli, R. S. Ingarden and M. Matsumoto, The theory of sprays and Finsler spaces with applications in physics and biology , Kluwer Acad. Publ., Dordrecht-Boston-London, 1993

    work page 1993

  3. [2]

    Bacs and M

    S. Bacs and M. Matsumoto, Reduction theorems of certain Landsberg spaces to Berwald s paces, Publ. Math. Debrecen 48 / 3-4 (1996), 357–366

    work page 1996

  4. [3]

    Berwald, On Finsler and Cartan geometries

    L. Berwald, On Finsler and Cartan geometries. III Two-dimensional Fins ler spaces with recti- linear extremals, Annals Math., 42 (1941), 84-112

    work page 1941

  5. [4]

    Guo and and X

    E. Guo and and X. Mo, The geometry of spherically symmetric Finsler manifolds . Springer Singapore, (2018)

    work page 2018

  6. [5]

    S. G. Elgendi, Finsler surfaces with vanishing T -tensor , J. Geom. Phys., 198 (2024), 105110

    work page 2024

  7. [6]

    S. G. Elgendi, On the problem of non-Berwaldian Landsberg spaces , Bull. Aust. Math. Soc., 102, (2020), 331–341

    work page 2020

  8. [7]

    S. G. Elgendi, Solutions for the Landsberg unicorn problem in Finsler geom etry, J. Geom. Phys., 159, (2021)

    work page 2021

Show all 24 references
  1. [8]

    S. G. Elgendi and L. Kozma, ( α, β)-metrics satisfying T-condition or σT-condition, J. Geom. Anal. (2020). 20

    2020

  2. [9]

    Hashiguchi, On conformal transformations of Finsler metrics , J

    M. Hashiguchi, On conformal transformations of Finsler metrics , J. Math. Kyoto Univ., 16 (1976), 25–50

    1976

  3. [10]

    M. A. Javaloyes and L. B. Soares, Geodesics and Jacobi fields of pseudo-Finsler manifolds , Publ. Math. Debrecen, 87 (2014), 57-78

    2014

  4. [11]

    M. A. Javaloyes and L. B. Soares, Anisotropic conformal invariance of lightlike geodesics i n pseudo-Finsler manifolds , Class. Quantum Grav., 38 (2021), 025002-025017

    2021

  5. [12]

    M. S. Knebelman, Conformal geometry of generalized metric spaces , Proc. Nat, Acad. Sci. USA, 15 (1929), 376-379

    1929

  6. [13]

    Ichijyo and M

    Y. Ichijyo and M. Hashiguchi, On the condition that a Randers space be conformally flat , Rep. Fac. Sci. Kagoshima Univ. 22 (1989), 7-14

    1989

  7. [14]

    Li and Z

    X. Li and Z. Chang, Exact solution of vacuum field equation in Finsler spacetime , Phys. Rev. D, 90 (6) (2014), 064049

    2014

  8. [15]

    Matsumoto, Finsler Geometry in the 20th-Century, Handbook of Finsler G eometry, Kluwer Academic Publishers, Dordrecht-Boston-London, 2003

    M. Matsumoto, Finsler Geometry in the 20th-Century, Handbook of Finsler G eometry, Kluwer Academic Publishers, Dordrecht-Boston-London, 2003

    2003

  9. [16]

    Nekouee, S

    Z. Nekouee, S. K. Narasimhamurthy and S.K.J. Pacif, Black hole solutions with constant Ricci scalar in a model of Finsler gravity , J. Cosmol. Astropart., 4 (2024), 061

    2024

  10. [17]

    Shen, S-closed conformal transformations in Finsler geometry , Diff

    B. Shen, S-closed conformal transformations in Finsler geometry , Diff. Geom. Appl. 58 (2018), 254-263

    2018

  11. [18]

    Shen and Z

    Y. Shen and Z. Shen, Introduction to modern Finsler geometry , World Scientific Publishing Co., Singapore, 2016

    2016

  12. [19]

    Shen, Two-dimensional Finsler metrics with constant flag curvatu re

    Z. Shen, Two-dimensional Finsler metrics with constant flag curvatu re. Manuscripta Math. 109, (2002), 349–366

    2002

  13. [20]

    Tachibana, On Finsler spaces which admit a concurrent vector field , Tensor N.S

    S. Tachibana, On Finsler spaces which admit a concurrent vector field , Tensor N.S. 1 (1950) 1–5

    1950

  14. [21]

    Yang and X

    G. Yang and X. Cheng, Conformal invariances of two-dimensional Finsler spaces w ith isotropic main scalar , Publ. Math. Debrecen 81 (3-4), (2012), 327–340

    2012

  15. [22]

    N. L. Youssef, S. G. Elgendi, A. A. Kotb and E. H. Taha, Anisotropic Conformal Change of Conic pseudo-Finsler Surfaces I , Class. Quantum Grav., 41 (2024), 175005

    2024

  16. [24]

    N. L. Youssef, S. G. Elgendi and E. H. Taha, Semi-Concurrent vector fields in Finsler geom- etry, Diff. Geom. Appl., 65 (2019), (1-15 pages). 21

    2019

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.